We insert the expressions (REF ), (REF ), and (REF ) for the
polarization vectors in (REF ).
All the Lorentz algebra is carried out using FORM [1]}.
Let us emphasize again that, in order to avoid possible ambiguities in the definition and application
of these external projectors, all pairs of Levi-Civita tensors in eq. (REF )
are replaced according to the contraction rule eq. (REF )
before being used in the projection. Then the projectors (REF ) are expressed
solely in terms of external momenta and space-time metric tensors.
After pulling out the normalization factors as prescribed in section (),
the resulting tensor projectors (which have only a polynomial dependence on external momenta and kinematics)
will be applied to the color stripped amplitudes
\(\mathcal {A}^{[\text{tree}]}_{gggg}(i), ~\mathcal {A}^{[\text{1-loop}]}_{gggg} (i)\) .
We use the convention to set the variable \(D=4\)  in the projectors (REF ),
in particular in the normalization factors that are pulled out. Of course this convention is used both
for the amplitudes and the associated UV and/or IR subtraction terms.
Then the normalization factors pulled out from the respective projectors (REF ) are
\(&&\mathcal {N}_{XXTT} = \frac{1}{s^2 t^2 (s + t)^2} ~,~\mathcal {N}_{XXYY} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\nonumber \\&&\mathcal {N}_{XYTY} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\mathcal {N}_{XYYT} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\nonumber \\&&\mathcal {N}_{YXTY} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\mathcal {N}_{YXTX} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\nonumber \\&&\mathcal {N}_{YYTT} = \frac{4}{s^2 t^2 (s + t)^2} ~,~\mathcal {N}_{YYYY} = \frac{16}{s^2 t^2 (s + t)^2} ~.\) 
